### Another H-super magic decompositions of the lexicographic product of graphs

#### Abstract

Let *H* and *G* be two simple graphs. The concept of an *H*-magic decomposition of *G* arises from the combination between graph decomposition and graph labeling. A decomposition of a graph *G* into isomorphic copies of a graph *H* is *H*-magic if there is a bijection *f* : *V*(*G*) ∪ *E*(*G*) → {1, 2, ..., ∣*V*(*G*) ∪ *E*(*G*)∣} such that the sum of labels of edges and vertices of each copy of *H* in the decomposition is constant. A lexicographic product of two graphs *G*_{1} and *G*_{2}, denoted by *G*_{1}[*G*_{2}], is a graph which arises from *G*_{1} by replacing each vertex of *G*_{1} by a copy of the *G*_{2} and each edge of *G*_{1} by all edges of the complete bipartite graph *K*_{n, n} where *n* is the order of *G*_{2}. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where *n* > 3, *m* > 1, and *t* = 3, 4, *n* − 2.

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DOI: http://dx.doi.org/10.19184/ijc.2018.2.2.2

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